Minimax Optimization: 
Minimax optimization is commonly employed to make learned models more robust, such as in distributionally robust optimization (DRO) and adversarially robust learning. In DRO, [1]} and [2]} showed that minimizing the maximum loss improves generalization due to a variance regularization effect. [3]} considers a robust model over multiple domains, showing convergence rates for both convex and non-convex losses. In MTL, DRO is shown to have strong generalization guarantees for unseen tasks in  [4]}, providing a theoretical motivation for robust MTL. In the context of adversarial learning and robustness, we also consider a minimax problem, where the minimum is over the loss and the maximum is over the worst-case perturbation to inputs [5]}, [6]}. [7]} customizes the perturbation level for each training sample through a minimax objective. In both DRO and robust learning, the minimax objective is in the form \(\min _x \max _y f(x, y)\) . Observe that this is a special case of the single-objective BO with independent lower and upper levels. In contrast, we study a much more general multi-objective BO problem where the upper level objective is dependent on the solution to the lower level objective. One key work that we highlight once more is Task-Robust MAML (TR-MAML) [8]}, which provides a robust meta-learning algorithm for few-shot tasks. They use a minimax objective (REF ), posing their optimization objective as a min-max single level multi-objective problem. In contrast, we focus on the min-max bilevel multi-objective BO problem.
